使用python实现前馈神经网络近似连续函数

下面是使用python实现前馈神经网络近似连续函数的示例代码：

import numpy as np
import matplotlib.pyplot as plt

# 定义输入数据
x = np.linspace(-10, 10, 1000)
y = np.sin(x)

# 定义神经网络
class NeuralNetwork:
    def __init__(self):
        # 随机生成权重和偏置
        self.weights1 = np.random.randn(1, 10)
        self.bias1 = np.random.randn(1, 10)
        self.weights2 = np.random.randn(10, 1)
        self.bias2 = np.random.randn(1, 1)
    
    def forward(self, x):
        # 前向传播
        self.layer1 = np.dot(x, self.weights1) + self.bias1
        self.activation1 = np.tanh(self.layer1)
        self.layer2 = np.dot(self.activation1, self.weights2) + self.bias2
        self.activation2 = self.layer2
        return self.activation2
    
    def backward(self, x, y, output, learning_rate):
        # 反向传播
        error = output - y
        d_layer2 = error
        d_weights2 = np.dot(self.activation1.T, d_layer2)
        d_bias2 = np.sum(d_layer2, axis=0, keepdims=True)
        d_activation1 = np.dot(d_layer2, self.weights2.T)
        d_layer1 = d_activation1 * (1 - np.power(self.activation1, 2))
        d_weights1 = np.dot(x.T, d_layer1)
        d_bias1 = np.sum(d_layer1, axis=0, keepdims=True)
        
        # 更新权重和偏置
        self.weights1 -= learning_rate * d_weights1
        self.bias1 -= learning_rate * d_bias1
        self.weights2 -= learning_rate * d_weights2
        self.bias2 -= learning_rate * d_bias2
    
    def train(self, x, y, learning_rate=0.1, epochs=1000):
        # 训练神经网络
        for i in range(epochs):
            output = self.forward(x)
            self.backward(x, y, output, learning_rate)
            if i % 100 == 0:
                loss = np.mean(np.square(output - y))
                print("Epoch:", i, "Loss:", loss)

# 创建神经网络实例
nn = NeuralNetwork()

# 训练神经网络
nn.train(x.reshape(-1, 1), y.reshape(-1, 1))

# 绘制原始数据和神经网络预测结果的比较图
plt.plot(x, y, label="Ground Truth")
plt.plot(x, nn.forward(x.reshape(-1, 1)).flatten(), label="Neural Network")
plt.legend()
plt.show()

上述代码首先定义了输入数据 `x` 和 `y`，其中 `y` 是 `x` 的正弦函数值。然后定义了一个名为 `NeuralNetwork` 的神经网络类，包含了前向传播、反向传播和训练方法。在训练方法中，通过循环迭代来更新权重和偏置，最终得到神经网络模型。最后，将原始数据和神经网络预测结果的比较图绘制出来。